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Three‐Phase Barker Arrays
Author(s) -
Bell Jason P.,
Jedwab Jonathan,
Khatirinejad Mahdad,
Schmidt KaiUwe
Publication year - 2015
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21377
Subject(s) - mathematics , aperiodic graph , phase (matter) , algebraic number , function (biology) , combinatorics , existential quantification , matrix (chemical analysis) , generating function , discrete mathematics , arithmetic , mathematical analysis , quantum mechanics , physics , materials science , evolutionary biology , composite material , biology
A 3‐phase Barker array is a matrix of third roots of unity for which all out‐of‐phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two‐dimensional 3‐phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3‐phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double‐exponentially growing arithmetic function T such that no 3‐phase Barker array of size s × t with 3 ∣ t exists for all t < T ( s ) . For example, T ( 5 ) = 4860 , T ( 10 ) > 10 11 , and T ( 20 ) > 10 214 . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3‐phase Barker array of size 3 r × 3 q exists, then r = q = 1 .